Geometry unit 9.5

UNIT 9.5 CONGRUENCEUNIT 9.5 CONGRUENCE

TRANSFORMATIONSTRANSFORMATIONS

Warm UpA figure has vertices A, B, and C. After a transformation, the image of the figure has vertices A′, B′, and C′. Draw the pre-image and the image on graph paper. Then identify the transformation.

1. A(-3, 1), B(-1, 1), C(-3, 4)

A′(3, 1), B′(5, 1), C′(3, 4)

translation 6 units right

2. A(2, 1), B(5, 1), C(4, 3)

A′(2, -1), B′(5, -1), C′(4, -3)

reflection across x-axis

Draw, identify, and describe transformations in the coordinate plane.

Use properties of rigid motions to determine whether figures are congruent and to prove figures congruent.

Objectives

dilation

isometry

rigid transformation

Vocabulary

A dilation with scale factor k > 0 and

center (0, 0) maps (x, y) to (kx, ky).

In a transformation, the original figure is the pre-image. The resulting figure is the image.

Remember!

Example 1: Drawing and Identifying Transformations

A. M: (x, y) (x - 4, y + 1)→

P(1, 3), Q(1, 1), R(4, 1)

translation 4 units left and 1 unit up

Apply the transformation M to the polygon with the given vertices. Identify and describe the transformation.

Example 1: Continued

B. M: (x, y) (x, -y)→

A(1, 2), B(4, 2), C(3, 1)

reflection across x-axis

Example 1: Continued

C. M: (x, y) (y, -x)→

R(-3, 0), E(-3, 3), C(-1, 3), T(-1, 0)

90°rotation clockwise with center of rotation (0, 0)

Example 1: Continued

D. M: (x, y) (3x, 3y)→

K(-2, -1), L(1, -1), N(1, -2))

dilation with scale factor 3 and center (0, 0)

Check It Out! Example 1

1. Apply the transformation M : (x, y) (3x, 3y) →to the polygon with vertices D(1, 3), E(1, -2), and F(3, 0). Name the coordinates of the image points. Identify and describe the transformation.

D’(3, 9), E’(3, -6), F’(9, 0); dilation with scale factor 3

An isometry is a transformation that preserves length, angle measure, and area. Because of these properties, an isometry produces an image that is congruent to the preimage.

A rigid transformation is another name for an isometry.

A. A(-3, 1), B(2, 3), C(1, 1)

P(-4, -2), Q(1, 0), R(0, -2)

Example 2: Determining Whether Figures are Congruent

The triangle are congruent; ABC can be mapped △to PQR by a translation: (x, y) (x - 1, y - 3).△ →

Determine whether the polygons with the given vertices are congruent.

B. A(2, -2), B(4, -2), C(4, -4)

P(3, -3), Q(6, -3), R(6, -6).

Example 2: Continued

The triangles are not congruent; ABC can be △mapped to PQR by a dilation with scale factor k ≠ △1: (x, y) (1.5x, 1.5y).→

Determine whether the polygons with the given vertices are congruent. Support your answer by describing a transformation: A(2, -1), B(3, 0), C(2, 3) and P(1, 2), Q(0, 3), R(-3, 2).

Check It Out! Example 2

The triangles are congruent because ABC can be mapped to PQR by a rotation: (x, y) (-y, x).→

Example 3: Applying Transformations

A(1, 2), B(2, 1), C(4, 2)

P(-3, -2), Q(-2, -1), R(-3, 1)

△ ABC can be mapped to A′B′C′ by a translation: △(x, y) (x – 3, y + 1); and then A′B′C′ can be → △mapped to PQR by a rotation: (x, y) (–y, x).△ →

Prove that the polygons with the given vertices are congruent.

Prove that the polygons with the given vertices are congruent: A(-4, -2), B(-2, 1), C( 2, -2) and P(1, 0), Q(3, -3), R(3, 0).

Check It Out! Example 3

The triangles are congruent because ABC can be mapped to A’B’C’ by a translation (x, y) (x + 5, y + 2); and then A’B’C’ →can be mapped to ABC by a reflection across the x-axis

Translations, reflections, and rotations can be called congruence transformations.

Helpful Hint

Example 4 : Architecture Application

Is there another transformation that can be used to create this frieze pattern? Explain your answer.

Example 4 : Continued

Repeated reflections can create this frieze pattern; a reflection of any section over a line through either the left or right side of each section.

Sketch a frieze pattern that can be produced by using reflections

Check It Out! Example 4

Possible answer: repeated horizontal reflections

Lesson Quiz : Part-IApply the transformation M to the polygon with the given vertices. Identify and describe the transformation.

dilation with scale factor 3 and center (0, 0)

1. M: (x, y) (3x, 3y)→

A(0, 1), B(2, 1), C(2, -1)

2. M: (x, y) (-y, x)→

A(0, 3), B(1, 2), C(4, 5)

90° rotation counterclockwise with center of rotation (0, 0)

Lesson Quiz : Part-II

translation 1 unit right and 2 units down

3. M: (x, y) (x + 1, y - 2)→

A(-2, 1), B(-2, 4), C(0, 3)

4. Determine whether the triangles are congruent. A(1, 1), B(1, -2), C(3, 0) J(2, 2), K(2, -4), L(6, 0)

not ; ABC can be mapped to JKL by a ≌ △ △dilation with scale factor k ≠ 1: (x, y) (2x, →2y).

Lesson Quiz : Part-III

△ ABC can be mapped to A′B′C′ by a △translation: (x, y) (x + 1, y + 4); and then → A′B′C′ can be mapped to DEF by a △ △

reflection: (x, y) (-x, y).→

5. Prove that the triangles are congruent. A(1, -2), B(4, -2), C(1, -4) D(-2, 2), E(-5, 2), F(-2, 0)

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